Statistical Methods for Spatio-Temporal Systems

✍ Scribed by Barbel Finkenstadt, Leonhard Held

• Publisher: Chapman and Hall\/CRC
• Year: 2006
• Tongue: English
• Leaves: 289
• Series: Monographs on Statistics and Applied Probability 107
• Edition: 1
• Category: Library

✦ Synopsis

Statistical Methods for Spatio-Temporal Systems presents current statistical research issues on spatio-temporal data modeling and will promote advances in research and a greater understanding between the mechanistic and the statistical modeling communities.Contributed by leading researchers in the field, each self-contained chapter starts with an introduction of the topic and progresses to recent research results. Presenting specific examples of epidemic data of bovine tuberculosis, gastroenteric disease, and the U.K. foot-and-mouth outbreak, the first chapter uses stochastic models, such as point process models, to provide the probabilistic backbone that facilitates statistical inference from data. The next chapter discusses the critical issue of modeling random growth objects in diverse biological systems, such as bacteria colonies, tumors, and plant populations. The subsequent chapter examines data transformation tools using examples from ecology and air quality data, followed by a chapter on space-time covariance functions. The contributors then describe stochastic and statistical models that are used to generate simulated rainfall sequences for hydrological use, such as flood risk assessment. The final chapter explores Gaussian Markov random field specifications and Bayesian computational inference via Gibbs sampling and Markov chain Monte Carlo, illustrating the methods with a variety of data examples, such as temperature surfaces, dioxin concentrations, ozone concentrations, and a well-established deterministic dynamical weather model.

✦ Table of Contents

Statistical Methods for Spatio-Temporal Systems......Page 5
Preface......Page 7
Participants......Page 10
Contributors......Page 13
Contents......Page 15
Contents......Page 16
1.1.1.1 Amacrine cells in the retina of a rabbit......Page 17
1.1.1.2 Bovine tuberculosis in Cornwall, U.K.......Page 18
1.1.1.3 Gastroenteric disease in Hampshire, U.K.......Page 19
1.1.1.4 The U.K. 2001 epidemic of foot-and-mouth disease......Page 20
1.1.2 Chapter outline......Page 21
1.2.1 Descriptors of pattern: spatial regularity, complete spatial randomness, and spatial aggregation......Page 22
1.2.2 Functional summary statistics......Page 23
1.2.3 Functional summary statistics for the amacrines data......Page 27
1.2.4.1 Pairwise interaction point processes......Page 28
1.2.4.2 Maximum pseudo-likelihood......Page 29
1.2.4.3 Monte Carlo maximum likelihood......Page 30
1.2.5 Bivariate pairwise interaction point processes......Page 31
1.2.6 Likelihood-based analysis of the amacrine cell data......Page 32
1.3 Strategies for the analysis of spatio-temporal point patterns......Page 33
1.3.1.2 A transition model for spatial aggregation......Page 34
1.3.1.3 Marked point process models......Page 35
1.3.2.1 Empirical modelling: log-Gaussian spatio-temporal Cox processes......Page 36
1.3.2.2 Mechanistic modelling: conditional intensity and a partial likelihood......Page 38
1.4 Bovine tuberculosis: non-parametric smoothing methods for estimating spatial segregation......Page 40
1.5 Real-time surveillance for gastroenteric disease: log-Gaussian Cox process modelling......Page 45
1.6 Foot-and-mouth disease: mechanistic modelling and partial likelihood analysis......Page 52
1.7 Discussion......Page 56
Acknowledgments......Page 57
References......Page 58
2.1 Introduction......Page 61
2.2.1 Setup......Page 64
2.2.2 The Poisson case......Page 66
2.2.3 Cox processes......Page 68
2.2.4.1 Inhomogeneous spatial point processes......Page 70
2.2.4.2 Spatio-temporal extensions......Page 73
2.3.1 Setup......Page 77
2.3.2 Levy-based growth models......Page 80
2.3.2.1 Linear Levy growth models......Page 81
2.3.2.2 Exponential Levy growth models......Page 83
Appendix A: Conditional densities and conditional intensities......Page 85
References......Page 87
Contents......Page 90
3.1 Introduction......Page 91
3.2.1.1 Continuous Fourier transform......Page 92
3.2.1.2.1 Scaling property......Page 93
3.2.1.2.3 Convolution theorem......Page 94
3.2.1.3 Aliasing......Page 95
3.2.2.1 The spectral representation theorem......Page 96
3.2.2.1.2 Bochner’s theorem......Page 97
3.2.3.1 Triangular model......Page 98
3.2.3.4 Mat´ern-Whittle class......Page 99
3.2.4.1 Periodogram......Page 103
3.2.4.2 Theoretical properties of the periodogram......Page 104
3.2.4.2.1 Asymptotic properties of the periodogram......Page 105
3.2.4.3.2 A forestry example......Page 106
3.2.4.5 Likelihood estimation in the spectral domain......Page 109
3.2.5.1 Convolution of locally stationary processes......Page 110
3.2.5.2 The spectrum for the convolution model......Page 111
3.2.5.4 Parametric spectral estimation......Page 113
3.2.5.5 An example in air quality......Page 114
3.2.6.1 Nonparametric estimation of a spatial spectrum......Page 117
3.2.6.2 Testing for stationarity......Page 119
3.2.6.3 An example in air quality......Page 121
3.2.6.3.1 Power of the test......Page 123
3.3 Wavelet analysis......Page 124
3.3.1 The continuous wavelet transform......Page 125
3.3.2 The discrete wavelet transform......Page 126
3.3.3.1 Mexican hat......Page 127
3.3.4.1 Fractional di.erence model for long-term memory......Page 129
3.3.4.2 The trend estimator......Page 130
3.3.4.4 Computing simultaneous con.dence band for the trend estimator......Page 131
3.3.4.5 Some spatial wavelets......Page 132
3.3.5 A nonstationary covariance structure......Page 134
3.4.1 Empirical orthogonal function analysis......Page 136
3.4.1.1 Computation of temporal trend basis functions from incomplete data using an SVD......Page 137
3.4.1.2 Spatial deformation modeling of the nonstationary spatial covariance structure of the detrended space-time residuals......Page 138
3.4.1.3 Example: Application to 8-hour maximum average daily ozone concentrations from southern California......Page 139
3.4.2.1 A spectral representation......Page 143
3.4.2.2 A new class of nonseparable space-time covariances......Page 145
3.4.2.3 An example in meteorology......Page 150
3.4.2.4 Testing for separability......Page 153
3.4.2.4.1 Test for separability......Page 154
3.4.2.5 An example in air quality......Page 156
3.5.1 What are the tools good for?......Page 158
3.5.3 Some research questions......Page 159
References......Page 160
4.1 Introduction......Page 163
4.2.1 Spatio-temporal domains......Page 165
4.2.2 Stationarity, separability, and full symmetry......Page 166
4.2.3 Positive definiteness......Page 168
4.3.1 Bochner’s Theorem......Page 169
4.3.2 Cressie-Huang representation......Page 171
4.3.3 Fully symmetric, stationary covariance functions......Page 172
4.3.4 Stationary covariance functions that are not fully symmetric......Page 173
4.3.5 Taylor’s hypothesis......Page 174
4.4 Irish wind data......Page 175
4.4.1 Exploratory analysis......Page 176
4.4.2 Fitting a parametric, stationary space-time correlation model......Page 179
4.4.3 Predictive performance......Page 180
References......Page 184
Contents......Page 188
5.1 Introduction......Page 189
5.2 Rainfall data and measurement......Page 190
5.2.1 Measurement using rain gauges......Page 191
5.2.2 Measurement using radar......Page 192
5.2.3 Data used in the examples......Page 193
5.3.1.1 Model de.nition and properties......Page 195
5.3.1.2 Model .tting......Page 198
5.3.1.3 Assessment of the .tted model......Page 201
5.3.2 Durations of events and dry periods......Page 202
5.3.3 Continuous simulation of a sequence of rain events......Page 206
5.4.1 Model construction......Page 209
5.4.2 Likelihood inference for time series data......Page 210
5.4.3 Inference for GLMs in a space-time setting......Page 212
5.4.4 Multi-site simulation......Page 214
5.4.5 Performance assessment......Page 216
5.5 Continuous simulations that are nonstationary in space and time......Page 219
5.5.2 Incorporating temporal nonstationarity......Page 220
5.5.3 Alternative approaches......Page 221
Acknowledgments......Page 222
References......Page 223
Contents......Page 227
6.2 Gaussian computation......Page 228
6.2.1 Generating multivariate normal realizations......Page 229
6.2.2 Conditional distributions......Page 233
6.2.3 Soft conditioning......Page 238
6.2.3.1 An example......Page 240
6.3 Gaussian Markov random .elds and Bayesian computation......Page 241
6.3.1 Locally linear Gaussian MRFs......Page 243
6.3.2 General Gaussian MRFs......Page 246
6.3.3.1 Iterated conditional modes......Page 248
6.3.3.2 Gibbs sampling......Page 249
6.3.4 Accounting for unknown model parameters......Page 250
6.4 Convolution-based spatial models......Page 254
6.4.1 Modeling and estimation......Page 258
6.4.2 Using a MRF model for x......Page 259
6.4.3 A multiresolution example......Page 263
6.4.4 A binary spatial model......Page 265
6.4.4.1 Archeology application......Page 268
6.5.1 A space-time convolution model......Page 269
6.5.2 A spatial convolution of a temporally evolving latent process......Page 272
6.6.1 The L96 model......Page 277
6.6.2 A Gaussian process model......Page 279
6.6.3 Bayesian inverse formulation......Page 281
6.6.4 PDE-based MRF formulation......Page 283
References......Page 287